master thesis 2009 weak values in quantum …shikano/uncertainty_principle.pdfmaster thesis 2009...

Master Thesis 2009Weak Values in Quantum Measurement Theory

— Concepts and Applications —

Yutaka Shikano07M01099

Department of Physics,Tokyo Institute of Technology

Supervisor: Professor Akio Hosoya

October 26, 2009

Contents

A Uncertainty Relationships 1A.1 Uncertainty Principle and Uncertainty Relationship . . . . . . . . . . . . . 1A.2 Quantum Mean Square Error . . . . . . . . . . . . . . . . . . . . . . . . . 3A.3 Detection Limit of Gravitational Wave . . . . . . . . . . . . . . . . . . . . 4A.4 Heisenberg’s Uncertainty Principle Revisited . . . . . . . . . . . . . . . . . 5A.5 Ozawa’s Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6A.6 Uncertainty Relationships for Joint Measurement . . . . . . . . . . . . . . 9

B Time-Energy Uncertainty Relationships 13

Bibliography 21

i

Appendix A

Uncertainty Relationships

In this chapter, we review various uncertainty relationships from the historical view1. Thishistorical review consists of the derivations and the definitions and the physical meanings.

A.1 Uncertainty Principle and Uncertainty Relation-

ship

The uncertainty principle was initiated by Heisenberg [55]. In the paper [55], Heisenbergexplained its physical meaning by the three examples, the gedankenexperiment of gamma-ray microscope gedankenexperiment (position-momentum), the Stern-Gerlach experiment(time-energy) (See App. B.), and the atomic structure (number-phase), and formulatedthese following the Dirac-Jordan theory [36, 75], which is the non-commutative theory. Inthe following, we explain the gamma-ray microscope gedankenexperiment (Fig. A.1) [56].The limits on the accuracy of the location 𝛿𝑥 of the image is given by

𝛿𝑥 ∼ 𝜆

sin 𝜖, (A.1)

where 𝜆 denotes the wave length of the scattered radiation2 and 𝜖 denotes the half angleof aperture of the object. The direction of the scattered light must then, in principle,be considered as undetermined within this angle 𝜖. Hence, according to the Comptoneffect, the component of the momentum of the material particle in the 𝑥-direction isundetermined, after the collision, by an amount

𝛿𝑝𝑥 ∼ ℎ

𝜆sin 𝜖. (A.2)

1As far as the author knows, there is no inclusive historical review of the uncertain relationships. Overthe historical review part, we refer to [104, 69]

2𝜆 can be different from the wave length of the incident radiation.

1

2 APPENDIX A. UNCERTAINTY RELATIONSHIPS

Figure A.1: Heisenberg’s gamma-ray gedanken microscope.

From these equations, we obtain

𝛿𝑥 ⋅ 𝛿𝑝𝑥 ∼ ℎ. (A.3)

This equation3 was based on classical mechanics but he derived it by the Dirac-Jordantheory.

In the last paragraph of the paper [55], Heisenberg added to the pre-publication proof 4

that Bohr pointed out the direct connection between the uncertain principle and the wave-particle duality. Then, Bohr introduced the complementarity based on the foundationsof quantum mechanics. The concept of complementarity is stated that a single quantummechanical entity can either behave as a particle or as wave, but never simultaneously asboth [19].

In the same year to publish the Heisenberg paper, Kennard derived the followinginequality [78] as

𝜎(𝑥) ⋅ 𝜎(𝑝) ≥ ℏ2, (A.4)

where 𝜎(𝑥) and 𝜎(𝑝) are the standard deviations of the position and momentum. Heconsidered when measuring some quantity, the amount of this probabilistically changesand we only statistically know the average value as we repeat the measurement. Heconcluded that this inequality was taken as the Heisenberg uncertainty principle (A.3)5

3Almost all physicists interpret the Heisenberg uncertainty principle as Eq. (A.17) or seem to be atcross-purposes with it as Eq. (A.5).

4We can get this from the following website.http://osulibrary.oregonstate.edu/specialcollections/coll/pauling/bond/papers/corr155.1.html

5In the abstract of the original paper: Das Ergebnis kann dahin formuliert werden, dass der Fallen nurin der Hisenbsergschen Unbestimmtheitsrelation zwischen den Werten kanonisch konjugierter Variabelnbesteht.

A.2. QUANTUM MEAN SQUARE ERROR 3

and there occurs the error in classical mechanics on the measurement but the error is thetheoretically inevitable quantity in quantum mechanics.

Weyl derived the same inequality (A.4) following the Cauchy-Bunyakowski-Schwarz in-equality [142, Appendix 1]. Robertson derived the general inequality for non-commutativeobservables [118],

𝜎(𝐴) ⋅ 𝜎(𝐵) ≥ ∣⟨[𝐴,𝐵]⟩∣2

, (A.5)

which is called Robertson’s uncertainty relationship and where 𝜎(𝐴) and ⟨𝐴⟩ are thestandard deviation and the average value of the observable 𝐴, respectively. Furthermore,Schrodinger derived the following inequality [124];

(𝜎(𝐴) ⋅ 𝜎(𝐵))2 ≥∣∣∣∣⟨[𝐴,𝐵]⟩

2

∣∣∣∣2 + (⟨{𝐴,𝐵}⟩2

− ⟨𝐴⟩ ⋅ ⟨𝐵⟩)2

. (A.6)

Summing up Robertson and Schrodinger’s works, we conclude that

𝜎(𝐴) ⋅ 𝜎(𝐵) ≥ ∣⟨𝐴𝐵⟩ − ⟨𝐴⟩⟨𝐵⟩∣ ≥ ∣⟨[𝐴,𝐵]⟩∣2

. (A.7)

The first inequality is due to Schrodinger, and the second to Robertson, which are derivedas follows [86].

Since ⟨𝐴𝐵⟩ = Tr[√𝜌𝐴𝐵

√𝜌] is an inner product for 𝐴

√𝜌 and 𝐵

√𝜌 as the Hilbert-

Schmidt operators for the density (Hermitian) operator 𝜌, the Cauchy-Bunyakowski-Schwarz inequality gives

⟨𝐴2⟩ ⋅ ⟨𝐵2⟩ ≥ ∣⟨𝐴𝐵⟩∣2. (A.8)

Replacing 𝐴 by 𝐴 − ⟨𝐴⟩ and 𝐵 by 𝐵 − ⟨𝐵⟩, we obtain the first inequality. For theHermitian operators, we obtain

∣⟨𝐴𝐵⟩∣2 ≥ 1

4∣⟨{𝐴,𝐵}⟩+ ⟨[𝐴,𝐵]⟩∣2 = 1

4∣⟨{𝐴,𝐵}⟩∣2 + 1

4∣⟨[𝐴,𝐵]⟩∣2 . (A.9)

Deleting the anti-commutator yields the Robertson inequality.

A.2 Quantum Mean Square Error

In classical mechanics and general statistical theory, the definition of the error is almostall taken as the mean square error since Gauss showed its efficiency [46]. Analogously, weintroduce the quantum mean square error as follows.Definition A.1 (Error operator). For any observable 𝐴 on the target system ℋ𝑠, theerror operator for the probe ℋ𝑝 is

𝑁𝐴 := 𝑈(𝐼 ⊗𝑀)𝑈 † − 𝐴⊗ 𝐼 :=𝑀out − 𝐴in, (A.10)


where 𝑈 is some evolution operator for the combined system and𝑀 is the meter observablefor the probe ℋ𝑝. Furthermore,

𝑛𝐴 := Trℋ𝑝 𝑁𝐴 (A.11)

is called an induced error operator for the observable 𝐴.Definition A.2 (Quantum mean square error). For any state 𝜌 and observable 𝐴 for thetarget system ℋ𝑠, the quantum mean square error for the probe ℋ𝑝 is

𝜖(𝐴) := ⟨𝑁2𝐴⟩1/2 (A.12)

= Tr ∣(𝑀out − 𝐴in)2(𝜌⊗ 𝜉)∣1/2, (A.13)

where 𝜉 is the probe initial state.

A.3 Detection Limit of Gravitational Wave

The discussion on the detection limit of gravitational waves substantially contributes tounderstand the Heisenberg uncertainty relationship. A gravitational wave is a fluctuationin the curvature of space-time which propagates as a wave, traveling outward from amoving object or system of objects. A gravitational radiation is the energy transported bythese waves. The existence of the gravitational wave is proven from the Einstein equationby Einstein [38]. As an examples of systems which emit gravitational waves, there arebinary star systems, where the two stars in the binary are white dwarfs, neutron stars, andblack holes. Although gravitational radiation has not yet been directly detected, it hasbeen indirectly shown to exist by slowing the period of revolution of PSR B1913+16 by76 microseconds per a year by Hulse and Taylor [67]. Recently, we try to directly detectgravitational waves by the Mach-Zehnder interferometer, e.g. LIGO6 and TAMA300 7.A passing a gravitational wave then slightly stretches one arm as it shortens the other.Therefore, the interference pattern is changed. The problem is how accurate can we decidethe length of the arms.

To discuss the problem to measure the mirror, the mirror of the interferometer is takenas a free particle with the mass 𝑚 and a quantum stuff. At some time, 𝑡 = 0, we measurethe position of the mirror. We measure it again at 𝑡 = 𝜏 . Caves et al. [27] showed

[𝜎(𝑥(𝜏))]2 = [𝜎(𝑥(0))]2 +

(𝜎(𝑝(0))𝜏

𝑚

)2

≥ 2𝜎(𝑥(0)) ⋅ 𝜎(𝑝(0)) ⋅ 𝜏𝑚

≥ ℏ𝜏𝑚, (A.14)

6Laser Interferometer Gravitational-Wave Observatory.7The 300m Laser Interferometer Gravitational Wave Antenna.

A.4. HEISENBERG’S UNCERTAINTY PRINCIPLE REVISITED 5

which the bound is called the standard quantum limit (SQL). However, Yuen pointed outthat the above inequality was wrong and was proposed as

[𝜎(𝑥(𝜏))]2 = [𝜎(𝑥(0))]2 +

(𝜎(𝑝(0))𝜏

𝑚

)2

[𝜎(𝑥(0)) ⋅ 𝜎(𝑝(0)) + 𝜎(𝑝(0)) ⋅ 𝜎(𝑥(0))] 𝜏𝑚, (A.15)

using the contractive state [145] and where the third term is negative. While Caves showedthat we cannot create the contractive state in the von Neumann model [28], Ozawa showedthat the following interaction Hamiltonian gives the contractive state as the ground state;

𝐻𝑖𝑛𝑡 =𝜋𝐾

3√3

(2��⊗ 𝑃 − 2𝑝⊗ �� + ��𝑝⊗ 𝐼 − 𝐼 ⊗ ��𝑃

), (A.16)

where 𝐾 is a coupling constant, which is so large to ignore the individual Hamiltonians,and (��, 𝑝) and (��, 𝑃 ) are the position and momentum operators on the target systemand the probe, respectively [98]. Finally, Maddox judged that Ozawa finally showed thedetection limit of this problem and concluded the paper in the hope to explore a newuncertainty relationship8 in Nature [87].

A.4 Heisenberg’s Uncertainty Principle Revisited

In the previous subsections, we have shown the difference between the error on the mea-surement the standard derivation of wave packets. Then, Ozawa discussed the Heisenberguncertainty relationship [101] as

𝜖(𝐴)𝜂(𝐵) ≥ ∣⟨[𝐴,𝐵]⟩∣2

, (A.17)

where 𝜂(𝐵) is defined as follows.As analogy to the error (Sec. A.2), we define the quantum mean square disturbance

as follows.Definition A.3 (Disturbance operator). For any observable 𝐵 on the target system ℋ𝑠,the disturbance operator for the probe ℋ𝑠 is

𝐷𝐵 := 𝑈(𝐵 ⊗ 𝐼)𝑈 † −𝐵 ⊗ 𝐼 := 𝐵out −𝐵in, (A.18)

where 𝑈 is some evolution operator for the combined system ℋ𝑠 ⊗ℋ𝑝. Furthermore,

𝑑𝐵 := Trℋ𝑝 𝐷𝐵 (A.19)

is called an induced disturbance operator for the observable 𝐵.8In the original paper: It is also far from obvious how the particular example on which the conclusion

rests can be turned into a realistic measuring equipment that would allow those who design equipmentto exploit this recipe for beating SQL. But none of this will damp enthusiasm for overcoming what oftenseems an intolerable constraint on the freedom to design accurate measuring equipment.


Definition A.4 (Quantum mean square disturbance). For any state 𝜌 and observable 𝐵for the target system ℋ𝑠, the quantum mean square disturbance for the probe ℋ𝑝 is

𝜂(𝐴) := ⟨𝐷2𝐵⟩1/2 (A.20)

= Tr ∣(𝐵out −𝐵in)2(𝜌⊗ 𝜉)∣1/2, (A.21)

where 𝜉 is the probe initial state.

A.5 Ozawa’s Inequality

Ozawa derived the uncertainty relationship included in the error and disturbance on themeasurement and the standard derivation of wave packets as

𝜖(𝐴)𝜂(𝐵) + 𝜖(𝐴)𝜎(𝐵) + 𝜎(𝐴)𝜂(𝐵) ≥ ∣⟨[𝐴,𝐵]⟩∣2

. (A.22)

This inequality means to violate the Heisenberg uncertainty relationship (A.17) and iscalled the Ozawa inequality [101].

On deriving the Ozawa inequality, it is useful the following lemma.Lemma A.1. Let 𝐴 and 𝜌 be a target observable and an initial target state on ℋ𝑠 and(ℋ𝑝, 𝜎, 𝑈,𝑀) be a quadruplet about the measurement, the probe Hilbert space, the initialprobe state, the unitary operator on the combined system, and the meter observable. Weobtain

𝜖(𝐴)𝜂(𝐵) +∣⟨[𝑁𝐴, 𝐵

in]⟩∣2

+∣⟨[𝐴in, 𝐷𝐵]⟩∣

2≥ ∣⟨[𝐴,𝐵]⟩∣

2. (A.23)

Proof. From the definitions of the error and disturbance operators (A.10, A.18), we obtain

0 = [𝐴out, 𝐵out]

= [𝐴in +𝑁𝐴, 𝐵in +𝐷𝐵] (A.24)

to transform[𝑁𝐴, 𝐷𝐵] + [𝑁𝐴, 𝐵

in] + [𝐴in, 𝐷𝐵] = −[𝐴in, 𝐵in]. (A.25)

Taking the average value for the density operator 𝜌⊗ 𝜎 on the both sides, we follow thetriangle inequality to obtain

∣⟨[𝑁𝐴, 𝐷𝐵]⟩∣+ ∣⟨[𝑁𝐴, 𝐵in]⟩∣+ ∣⟨[𝐴in, 𝐷𝐵]⟩∣ ≥ ∣⟨[𝐴in, 𝐵in]⟩∣

= ⟨[𝐴,𝐵]⟩. (A.26)

Using the relationship as the above,

𝜖(𝐴) ≥ 𝜎(𝑁𝐴), (A.27)

𝜂(𝐵) ≥ 𝜎(𝐷𝐵), (A.28)

A.5. OZAWA’S INEQUALITY 7

and the Robertson uncertainty relationship (A.5), we obtain

𝜖(𝐴) ⋅ 𝜂(𝐵) ≥ 𝜎(𝑁𝐴) ⋅ 𝜎(𝐷𝐵)

≥ ⟨[𝑁𝐴, 𝐷𝐵]⟩2

. (A.29)

Then, we obtain

𝜖(𝐴)𝜂(𝐵) +∣⟨[𝑁𝐴, 𝐵

in]⟩∣2

+∣⟨[𝐴in, 𝐷𝐵]⟩∣

2≥ ⟨[𝑁𝐴, 𝐷𝐵]⟩

2+

∣⟨[𝑁𝐴, 𝐵in]⟩∣

2+

∣⟨[𝐴in, 𝐷𝐵]⟩∣2

≥ ∣⟨[𝐴,𝐵]⟩∣2

, (A.30)

following the second inequality (A.23), which is the desired inequality.From the left hand side of Eq. (A.22), we obtain

𝜖(𝐴)𝜂(𝐵) + 𝜖(𝐴)𝜎(𝐵) + 𝜎(𝐴)𝜂(𝐵)

≥ 𝜖(𝐴)𝜂(𝐵) + 𝜎(𝑁𝐴)𝜎(𝐵) + 𝜎(𝐴)𝜎(𝐷𝐵)

≥ 𝜖(𝐴)𝜂(𝐵) +

∣∣∣∣⟨[𝑁𝐴, 𝐵in]⟩

2+

⟨[𝐴in, 𝐷𝐵]⟩2

∣∣∣∣≥ 𝜖(𝐴)𝜂(𝐵) +

∣⟨[𝑁𝐴, 𝐵in]⟩∣

2+

∣⟨[𝐴in, 𝐷𝐵]⟩∣2

≥ ∣⟨[𝐴,𝐵]⟩∣2

, (A.31)

following Eqs. (A.10) and (A.18) in the first inequality, the Robertson uncertainty rela-tionship (A.5) in the second inequality, the triangle inequality in the third inequality, andEq. (A.26) in the forth inequality.

The case of holding the Heisenberg uncertainty relationship (A.17) is .

∣⟨[𝑁𝐴, 𝐵in]⟩∣+ ∣⟨[𝐴in, 𝐷𝐵]⟩∣ = 0. (A.32)

In order to characterize a class of measurements satisfying Eq. (A.17), we define that themeasurement interaction is said to be independent intervention for the pair (𝐴,𝐵) if thenoise and the disturbance are independent of the target system; or precisely if there isobservables 𝑁 and 𝐷 of the probe such that 𝑁𝐴 = 𝐼 ⊗ 𝑁 and 𝐷𝐵 = 𝐼 × 𝐵. Under thiscondition, we obtain

⟨[𝑁𝐴, 𝐵in]⟩ = ⟨[𝐴in, 𝐷𝐵]⟩ = 0. (A.33)

Therefore, the Ozawa inequality (A.22) reduces the Heisenberg uncertainty relation-ship (A.17). The above conclusion was previously suggested in part by Braginsky andKhalili [22, p. 65] with a limited justification and now fully justified9.

9Generally speaking, the condition that the Ozawa inequality reduces the Heisenberg uncertaintyrelationship is given by [103, Theorems 6.1 and 6.3].


We consider the case to violate the Heisenberg uncertainty relationship (A.17) tomeasure the position. We assume that the interaction Hamiltonian as the above (Sec.A.3) be

𝐻𝑖𝑛𝑡 =𝜋𝐾

3√3

(2��⊗ 𝑃 − 2𝑝⊗ �� + ��𝑝⊗ 𝐼 − 𝐼 ⊗ ��𝑃

), (A.34)

where 𝐾 is a coupling constant, which is so large to ignore the individual Hamiltonians,and (��, 𝑝) and (��, 𝑃 ) are the position and momentum operators on the target systemand the probe, respectively. Then, the evolution operator is given by

𝑈(𝑡) = exp

(−𝑖𝐻𝑖𝑛𝑡𝑡

ℏ

). (A.35)

We denote the initial probe state as ∣𝜉⟩ and the time to end the measurement interactionas 𝑡 = Δ𝑡. In the follows, we consider the Heisenberg picture. From the Heisenbergequation, we obtain

𝑑

𝑑𝑡��(𝑡) =

𝜋𝐾

3√3[��(𝑡)− 2��(𝑡)],

𝑑

𝑑𝑡𝑝(𝑡) = − 𝜋𝐾

3√3[𝑝(𝑡) + 2𝑃 (𝑡)],

𝑑

𝑑𝑡��(𝑡) =

𝜋𝐾

3√3[2��(𝑡)− ��(𝑡)],

𝑑

𝑑𝑡𝑃 (𝑡) =

𝜋𝐾

3√3[2𝑝(𝑡)− 𝑃 (𝑡)], (A.36)

to calculate ��(𝑡) and ��(𝑡) as

𝑑

𝑑𝑡(��(𝑡) + ��(𝑡)) =

𝜋𝐾√3[��(𝑡)− ��(𝑡)],

𝑑

𝑑𝑡(��(𝑡) + ��(𝑡)) = − 𝜋𝐾

3√3[��(𝑡) + ��(𝑡)]. (A.37)

We calculate𝑑2

𝑑𝑡2(��(𝑡) + ��(𝑡)) = −

(𝜋𝐾

3

)2

(��(𝑡) + ��(𝑡)) (A.38)

to obtain

��(𝑡) + ��(𝑡) = 𝐴 exp

(𝑖𝜋𝐾

3𝑡

)+ �� exp

(−𝑖𝜋𝐾

3𝑡

). (A.39)

From the initial condition, 𝑡 = 0, 𝐴 and �� are determined. By the analogous discussion,

A.6. UNCERTAINTY RELATIONSHIPS FOR JOINT MEASUREMENT 9

we obtain

��(𝑡) =2√3

(sin

(𝜋3(1 +𝐾𝑡)

)��(0)− sin

(𝜋3𝐾𝑡

)��(0)

)𝑝(𝑡) =

2√3

(sin

(𝜋3(1−𝐾𝑡)

)𝑝(0)− sin

(𝜋3𝐾𝑡

)𝑃 (0)

)��(𝑡) =

2√3

(sin

(𝜋3𝐾𝑡

)��(0) + sin

(𝜋3(1−𝐾𝑡)

)��(0)

)𝑃 (𝑡) =

2√3

(sin

(𝜋3𝐾𝑡

)𝑝(0) + sin

(𝜋3(1 +𝐾𝑡)

)𝑃 (0)

)(A.40)

Setting Δ𝑡 = 1/𝐾, we calculate⎛⎜⎜⎝��(Δ𝑡)𝑝(Δ𝑡)

��(Δ𝑡)

𝑃 (Δ𝑡)

⎞⎟⎟⎠ =

⎛⎜⎜⎝��(0)− ��(0)

−𝑃 (0)��(0)

𝑝(0) + 𝑃 (0)

⎞⎟⎟⎠ (A.41)

to obtain the error and disturbance operators as

𝑁𝑥 = ��(Δ𝑡)− ��(0) = 0, (A.42)

𝐷𝑝 = 𝑝(Δ𝑡)− 𝑝(0) = −(𝑝⊗ 𝐼 + 𝐼 ⊗ 𝑃

)(A.43)

. The quantum mean square error and disturbance are given by

𝜖(𝑥) = ⟨𝑁2𝑥⟩ = 0 (A.44)

𝜂(𝑝) = ⟨𝐷2𝑝⟩ = 𝜎(𝑝) + 𝜎(𝑃 ) + ∣⟨𝑝⟩+ ⟨𝑃 ⟩∣2 <∞. (A.45)

Therefore, the Heisenberg uncertainty relations (A.17) can be transformed as

𝜖(𝑥) ⋅ 𝜂(𝑝) = 0. (A.46)

This means that the Heisenberg uncertainty relations (A.17) can be violated.

A.6 Uncertainty Relationships for Joint Measurement

We consider joint measurement of non-commutative observables 𝐴 and 𝐵 under the un-biased condition, that is,

⟨𝐴⟩ = ⟨𝐴⟩meas, ⟨𝐵⟩ = ⟨𝐵⟩meas, (A.47)


hold for the all target states 𝜌. Ishikawa and Ozawa [70, 71, 100] independently derivedthe uncertainty relationship between quantum mean square errors as

𝜖(𝐴) ⋅ 𝜖(𝐵) ≥ ∣⟨[𝐴,𝐵]⟩∣2

, (A.48)

which is called the Ishikawa-Ozawa inequality10. They assumed that the probe consistsof macroscopic stuffs to satisfy

[𝑈(𝐼 ⊗𝑀𝐴)𝑈†, 𝑈(𝐼 ⊗𝑀𝐵)𝑈

†] = 0, (A.49)

where 𝑀𝐴 and 𝑀𝐵 are meter observables to measure the target observables 𝐴 and 𝐵,respectively. From Eq. (A.49), we obtain that

[𝑁𝐴, 𝑁𝐵] + [𝑁𝐴, 𝐵 ⊗ 𝐼] + [𝐴⊗ 𝐼,𝑁𝐵] + [𝐴,𝐵]⊗ 𝐼 = 0, (A.50)

where the error operator 𝑁𝐴 and 𝑁𝐵 are defined in Eq. (A.10). From the unbiasedcondition, we obtain

Tr(𝑛𝐴𝜌) = 0, (A.51)

for all 𝜌 to restrict the induced error operator as

𝑛𝐴 = 0. (A.52)

Then, we obtainTr([𝑁𝐴, 𝐵 ⊗ 𝐼](𝜌⊗ 𝜉)) = Tr([𝑛𝐴, 𝐵]𝜌) = 0, (A.53)

and similarly Tr([𝑁𝐵, 𝐴 ⊗ 𝐼](𝜌 ⊗ 𝜉)) = 0, where 𝜉 is the probe initial state. Taking theaverage of both sides of Eq. (A.50) in the state 𝜌⊗ 𝜉, we obtain

Tr([𝑁𝐴, 𝑁𝐵](𝜌⊗ 𝜉)) = −Tr([𝐴,𝐵]𝜌). (A.54)

Noting that(𝜎(𝑁𝐴))

2 = (𝜖(𝐴))2 − ∣Tr(𝑁𝐴(𝜌⊗ 𝜉))∣2 ≤ (𝜖(𝐴))2, (A.55)

we obtain

𝜖(𝐴) ⋅ 𝜖(𝐵) ≥ 𝜎(𝑁𝐴) ⋅ 𝜎(𝑁𝐵)

≥ ∣Tr([𝑁𝐴, 𝑁𝐵](𝜌⊗ 𝜉))∣2

=∣Tr([𝐴,𝐵]𝜌)∣

2, (A.56)

10Strictly speaking, the inequality derived by Ishikawa [70] is different by the meaning of the erroroperator.

A.6. UNCERTAINTY RELATIONSHIPS FOR JOINT MEASUREMENT 11

following the Robertson uncertainty relationship (A.5) in the second inequality, which isdesired inequality (A.48). We have applied this to various measurement model (e.g., see[121]).

From the view of the joint measurement, the Ozawa inequality [102] can be analogouslyderived as

𝜖(𝐴)𝜖(𝐵) + 𝜖(𝐴)𝜎(𝐵) + 𝜎(𝐴)𝜖(𝐵) ≥ ∣⟨[𝐴,𝐵]⟩∣2

. (A.57)

Under the unbiased condition, we have proven that this inequality is reduced to theIshikawa-Ozawa inequality. Therefore, the Ozawa inequality is substantial under the casewithout the unbiased condition.

Appendix B

Time-Energy UncertaintyRelationships

In this chapter, we review the papers on the time-energy uncertainty relationships.

Paper 1 (Heisenberg Uncertainty Principle [55]).

𝜖(𝐸)𝜂(𝑇 ) ∼ ℎ, (B.1)

where 𝜖(𝐸) is defined the error to measure the energy of the system and 𝜂(𝑇 ) is definedthe time to measure the energy. This relation always keeps when we measure the energyof a specified system.

Einstein posed a paradox during the sixth Solvay conference in 1930 to violate thetime-energy uncertainty relationship,

Δ𝐸Δ𝑡 ≥ ℎ, (B.2)

where Δ𝐸 and Δ𝑡 were defined as uncertainties of the measurement using a box thatemits a photon (See Fig. B.1). This is because the box hangs from a spring scale whichmeasures its weight. Its weight is proportional to its rest mass 𝑚, hence to its energy 𝐸,according to 𝐸 = 𝑚𝑐2. Einstein supposed that we wait for the box to settle down andaccurately measure the initial scale. This reading can be as slow and accurate as we like.After the photon leaves the box, we measure the final scale, again as accurate as we like,and from the difference between the two positions we get an accurate measurement of theenergy of the emitted photon, that is, Δ𝐸 <∞. On the other hand, the clock in the boxtells exactly when the photon was released, that is, Δ𝑡 = 0, so we violate Eq. (B.2)1.

1 Rosenfeld described the Bohr reaction to this argument. ”It was quite a shock for Bohr to be facedwith this problem; he did not see the solution at once. During the whole evening, he was extremelyunhappy, going from one to the other and trying to persuade them that it could not be true, that itwould be the end of physics if Einstein were right; but he could not produce any refutation... The nextmorning came Bohr’s triumph and the salvation of physics...” [105, P.238].

13

14 APPENDIX B. TIME-ENERGY UNCERTAINTY RELATIONSHIPS

Figure B.1: The clock model proposed by Bohr [20] when Bohr triumphed against Ein-stein’s criticism.

Bohr triumphed as follows. Let 𝑥 denote the position of the pointer on the scale and𝑝 its momentum, with Δ𝑥 and Δ𝑝 the corresponding uncertainties. Once we choose Δ𝑥,Δ𝑝 is restricted from the Heisenberg uncertainty relationship, Δ𝑥 ⋅Δ𝑝 ≥ ℎ, as

ℎ

Δ𝑥≤ Δ𝑝. (B.3)

Bohr assumed the pointer moves after the photon emission. By hanging little weights onthe box, we lower it to its original position. When it has returned to its original height,the total wight hanging from it equals the weight of the emitted photon. However, theaccurate of this weighting is no better than the smallest added weight 𝑔Δ𝑚 that has anobservable effect. If we add a mass Δ𝑚 and wait a time 𝑡, the impulsive delivered to thebox cannot be greater than (𝑔Δ𝑚)𝑡, which must be greater than Δ𝑝 to be observable toobtain

Δ𝑝 ≤ 𝑔𝑡Δ𝑚. (B.4)

From Eqs. (B.3) and (B.4), we obtain

ℎ ≤ 𝑔𝑡Δ𝑥Δ𝑚 =𝑔𝑡Δ𝑥Δ𝐸

𝑐2. (B.5)

Einstein assumed that to measure the pointer position could take unlimited time. ButBohr applied a result from general relativity. According to the time-dilation formula of

15

general relativity, a clock in gravitational field ticks more slowly than a clock in free fall.Two clocks at different heights above the Earth will run at different rates, because oftheir gravitational potential difference. If the difference in height is Δ𝑥, the fractionaldifference Δ𝑡/𝑡 in their measured times will be

Δ𝑡

𝑡=𝑔Δ𝑥

𝑐2. (B.6)

If Δ𝑥 is the uncertainty in the vertical position of a clock, then Δ𝑡 is the uncertainty in theclock time due to the uncertain gravitational potential. Over a period 𝑡, the uncertaintyin the time of the clock amounts to

Δ𝑡 =𝑡𝑔Δ𝑥

𝑐2. (B.7)

Combining this result with Eq. (B.5), we obtain

Δ𝑡Δ𝐸 ≥ ℎ, (B.8)

as required by quantum theory.

Paper 2 (Salecker and Wigner [122]). Let us consider a linear clock model.

Δ𝑡𝑇 >

√ℏ𝑇𝐸, (B.9)

where Δ𝑡𝑇 = Δ𝑥/⟨𝑣⟩ means that a clock measures a time after 𝑇 , i.e., this is an accuracy.We call the first Salecker-Wigner (S-W) inequality.

𝐸 >ℏ

2Δ𝑡𝑇

√𝑇

Δ𝑡𝑇. (B.10)

We call the second S-W inequality.

We derive the first S-W inequality as follows. From the Heisenberg uncertainty rela-tionship (A.5), we have Δ𝑥Δ𝑝 ≥ ℏ/2, where Δ𝑥 is a variance in the clock position. Weassume that the spread in velocity Δ𝑣 = Δ𝑝/𝑀 , where𝑀 is a mass of the clock, remains.Over time 𝑡, the variance of the position grows as

Δ𝑥2𝑡 = Δ𝑥20 + 𝑡2Δ𝑣2 = Δ𝑥20 +ℏ2𝑡2

4𝑀2Δ𝑥20. (B.11)

Fixing the overall time 𝑇 and minimizing over Δ𝑥0 yields a clock accuracy

Δ𝑥𝑇 ≥√

ℏ𝑇𝑀. (B.12)


Since ⟨𝑣⟩ < 𝑐, Eq. (B.12) can be transformed to the desired inequality.We derive the second S-W inequality as follows. The reading of the clock is connected

with the emission of a light signal of duration Δ𝑡𝑇 and this imparts to the clock anindeterminate momentum ℏ/𝑐Δ𝑡𝑇 . This momentum wold be even greater if a particle ofnonzero rest mass were used as a signal. As a result of the emission of the light signal, thevelocity of the clock acquires a spread of the amount ℏ/𝑀𝑐Δ𝑡𝑇 , where 𝑀 is the mass ofthe clock. After a further time interval 𝑇2, it may be at a distance ℏ𝑇2/𝑀𝑐Δ𝑡𝑇 from thepoint where it would have been without having been read. Therefore, the actual distancebetween the two points in space time, at the first of which the clock read 𝑇1 less than atthe time of the emission of the signal, at the second of which it reads 𝑇2 more than at thetime of the emission of the light signal, is√

(𝑇1 + 𝑇 ′2)

2 −(

ℏ𝑇2𝑀𝑐2Δ𝑡𝑇

)2

∼ 𝑇1 + 𝑇 ′2 −

ℏ2𝑇 22

2𝑀2𝑐4(Δ𝑡𝑇 )2(𝑇1 + 𝑇 ′2), (B.13)

where

𝑇 ′2 =

√𝑇 22 +

(ℏ𝑇2

𝑀𝑐2Δ𝑡𝑇

)2

. (B.14)

Hence, the actual distance (B.13) differs from the time difference 𝑇 = 𝑇1 + 𝑇2 shown bythe clock, in the approximation considered, by

− 𝑇1𝑇22(𝑇1 + 𝑇2)

(ℏ𝑇2

𝑀𝑐2Δ𝑡𝑇

)2

. (B.15)

The inaccuracy of the clock will be within the limit Δ𝑡𝑇 if Eq. (B.15) is less than Δ𝑡𝑇 .If one considers the first order to be of the order of magnitude 𝑇 , we obtain

𝑀 >ℏ

𝑐2Δ𝑡𝑇

√𝑇

Δ𝑡𝑇, (B.16)

which is corresponded to the desired equation.Summing up the two S-W inequalities, the first inequality is restricted from the con-

stant light speed 𝑐 and the second one is restricted from the causality.

Paper 3 (Margolus and Levitin [89]).

Δ𝑡 ≥ 𝜋ℏ2𝐸

, (B.17)

where Δ𝑡 is defined as the time which can move from one state to an orthogonal statewith a fixed average energy, denoted as 𝐸.

17

This relation is derived as the follows. An arbitrary quantum state can be written asa superposition of energy eigenstates

∣𝜓0⟩ =∑𝑛

𝑐𝑛∣𝐸𝑛⟩. (B.18)

Note that we assume that a system has a discrete spectrum {𝐸𝑛} and choose the groundenergy zero so that 𝐸0 = 0. If ∣𝜓0⟩ is evolved for a time 𝑡, then it becomes

∣𝜓𝑡⟩ =∑𝑛

𝑐𝑛𝑒−𝑖𝐸𝑛𝑡/ℏ∣𝐸𝑛⟩. (B.19)

In order to judge the orthogonality for these states, ∣𝜓0⟩ and ∣𝜓𝑡⟩, we let

𝑆(𝑡) ≡ ⟨𝜓0∣𝜓𝑡⟩ =∑𝑛

∣𝑐𝑛∣2𝑒−𝑖𝐸𝑛𝑡/ℏ. (B.20)

Since we want to solve the time which can move from one state to an orthogonal state,we want to find the smallest value of 𝑡 such that 𝑆(𝑡) = 0. To do this, we note that

Re(𝑆(𝑡)) =∑𝑛

∣𝑐𝑛∣2 cos(𝐸𝑛𝑡

ℏ

)≥

∑𝑛

∣𝑐𝑛∣2(1− 2

𝜋

(𝐸𝑛𝑡

ℏ+ sin

(𝐸𝑛𝑡

ℏ

)))= 1− 2𝐸

𝜋ℏ𝑡+

2

𝜋Im(𝑆(𝑡)), (B.21)

where we have used the inequality cos𝑥 ≥ 1 − (2/𝜋)(𝑥 + sin𝑥) for 𝑥 ≥ 0. On 𝑆(𝑡) = 0,both Re(𝑆(𝑡)) = 0 and Im(𝑆(𝑡)) = 0, and so Eq. (B.21) becomes

0 ≥ 1− 2𝐸

𝜋ℏ𝑡. (B.22)

Then we obtain the desired inequality.

Paper 4 (Anandan and Aharonov [14], Vaidman [138]).

Δ𝑡 ≥ 𝜋ℏ2Δ𝐸

, (B.23)

where Δ𝑡 is defined as the time which can move from one state to an orthogonal state andΔ𝐸 is defined as the variance of the energy.


In this thesis, Vaidman’s proof [138] is as follows. In general, for a given observable𝐴, we can decompose

𝐴∣𝜓⟩ = 𝛼∣𝜓⟩+ 𝛽∣𝜓⊥⟩, (B.24)

where ∣𝜓⊥⟩ is orthogonal to ∣𝜓⟩ and 𝛽 is real and non-negative. Then ⟨𝐴⟩ ≡ ⟨𝜓∣𝐴∣𝜓⟩ =⟨𝜓∣(𝛼∣𝜓⟩ + 𝛽∣𝜓⊥⟩) yields 𝛼 = ⟨𝐴⟩, and ⟨𝜓∣𝐴†𝐴∣𝜓⟩ = (𝛼∗∣𝜓⟩+ 𝛽∗∣𝜓⊥⟩) (𝛼∣𝜓⟩+ 𝛽∣𝜓⊥⟩)yields 𝛽 =

√⟨𝐴2⟩ − ⟨𝐴⟩2 ≡ Δ𝐴 to obtain

𝐴∣𝜓⟩ = ⟨𝐴⟩∣𝜓⟩+Δ𝐴∣𝜓⊥⟩. (B.25)

From this and the Schrodinger equation, we obtain

𝑑

𝑑𝑡∣𝜓(𝑡)⟩ = − 𝑖

ℏ𝐻∣𝜓(𝑡)⟩ = − 𝑖

ℏ(⟨𝐸⟩∣𝜓(𝑡)⟩+Δ𝐸∣𝜓⊥(𝑡)⟩) , (B.26)

where ⟨𝜓(𝑡)∣𝜓⊥(𝑡)⟩ = 0. Furthermore, we calculate

𝑑

𝑑𝑡∣⟨𝜓(𝑡)∣𝜓(0)⟩∣2 = 2Re

(⟨𝜓(𝑡)∣𝜓(0)⟩⟨𝜓(0)∣ 𝑑

𝑑𝑡∣𝜓(𝑡)⟩

). (B.27)

From these, we obtain

𝑑

𝑑𝑡∣⟨𝜓(𝑡)∣𝜓(0)⟩∣2 = −2Δ𝐸

ℏRe (𝑖⟨𝜓(𝑡)∣𝜓(0)⟩⟨𝜓(0)∣𝜓⊥(𝑡)⟩) . (B.28)

Furthermore, we expand the initial state ∣𝜓(0)⟩ as∣𝜓(0)⟩ = ⟨𝜓(𝑡)∣𝜓(0)⟩∣𝜓(𝑡)⟩+ ⟨𝜓⊥(𝑡)∣𝜓(0)⟩∣𝜓⊥(𝑡)⟩+ 𝛼∣𝜓⊥⊥(𝑡)⟩, (B.29)

where ⟨𝜓(𝑡)∣𝜓⊥⊥(𝑡)⟩ = 0 and ⟨𝜓⊥(𝑡)∣𝜓⊥⊥(𝑡)⟩ = 0. The normalization of the quantumstates, then, requires that

∣⟨𝜓(0)∣𝜓⊥(𝑡)⟩∣2 = 1− ∣⟨𝜓(𝑡)∣𝜓(0)⟩∣2 − ∣𝛼∣2. (B.30)

Therefore, the maximum value of ∣⟨𝜓(0)∣𝜓⊥(𝑡)⟩ is obtained for 𝛼 = 0, and it is equal to√1− ∣⟨𝜓(0)∣𝜓(𝑡)⟩∣2. Thus, the maximum possible absolute value of the rate of change of

the square of the overlap is

2Δ𝐸

ℏ∣⟨𝜓(0)∣𝜓(𝑡)⟩∣

√1− ∣⟨𝜓(0)∣𝜓(𝑡)⟩∣2. (B.31)

We find that this maximum rate, indeed, depends only on the value of the overlap and onthe energy uncertainty. Therefore, the condition for the fastest evolution to an orthogonalstate is that during the whole period of the evolution the right-hand side of Eq. (B.28) isequal to minus Eq. (B.31):

𝑑

𝑑𝑡∣⟨𝜓(𝑡)∣𝜓(0)⟩∣2 = −2Δ𝐸

ℏ∣⟨𝜓(0)∣𝜓(𝑡)⟩∣

√1− ∣⟨𝜓(0)∣𝜓(𝑡)⟩∣2, (B.32)

19

After introducing a parameter 𝜙, cos𝜙 = ∣⟨𝜓(0)∣𝜓(𝑡)⟩∣, Eq. (B.32) becomes

𝑑

𝑑𝑡𝜙 =

Δ𝐸

ℏ. (B.33)

Since the orthogonal state corresponds to 𝜙 = 𝜋/2, the minimum time is, indeed

Δ𝑡 =𝜋

2��=

ℎ

4Δ𝐸. (B.34)

The relationship is desired.

Paper 5 (Mandelstam and Tamm [88]).

Δ𝐻Δ𝑇 ≥ ℏ2, (B.35)

where Δ𝐻 is defined as an uncertainty of the energy and Δ𝑇 is defined as the shortesttime, during which the average value of a certain quantity is changed by an amount equalto the standard deviation of this quantity.

This equation is derived as the follows. Let 𝑅 and 𝑆 denote any two physical quantitiesand at the same time the corresponding symmetric operators. We have shown that usingthe Cauchy-Schwartz inequality,

Δ𝑆Δ𝑅 ≥ 1

2∣⟨𝑅𝑆 − 𝑆𝑅⟩∣, (B.36)

where Δ𝑆 and Δ𝑅 are the standard deviations of the quantities 𝑆 and 𝑅 and ⟨⋅⟩ denotesas the average value, and the Heisenberg equation as

ℏ∂⟨𝑅⟩∂𝑡

= 𝑖 (⟨𝐻𝑅−𝑅𝐻⟩) . (B.37)

Putting in (B.36) 𝑆 ≡ 𝐻, we obtain the inequality,

Δ𝐻Δ𝑅 ≥ ℏ2

∣∣∣∣∂⟨𝑅⟩∂𝑡

∣∣∣∣ . (B.38)

The absolute value of an integral cannot exceed th integral of the absolute value of theintegrand. Hence, integrating (B.38) from 𝑡 to 𝑡 + Δ𝑡 and taking into account that Δ𝐻is constant one gets

Δ𝐻Δ𝑡 ≥ ℏ2

∣⟨𝑅𝑡+Δ𝑡⟩ − ⟨𝑅𝑡⟩∣Δ⟨𝑅⟩ . (B.39)

This relation is desired.


Paper 6 (Aharonov and Bohm [4]).

Δ𝐻Δ𝑇 ≥ ℏ2, (B.40)

where Δ𝐻 is defined as an uncertainty of the energy and Δ𝑇 is defined as the shortesttime, during which the average value of a certain quantity is changed by an amount equalto the standard deviation of this quantity. Aharonov and Bohm basically showed thatMandelstam and Tamm bound is universal as follows.

They define the ”clock” operator,

𝑇𝑐 =1

2

{��,

1

𝑝

}, (B.41)

noting that this operator is singular. When the ”clock” Hamiltonian is given ��𝑐 = 𝑝2/2𝑀 ,we obtain the commutation relations,

[��𝑐, 𝑇𝑐] = 𝑖ℏ. (B.42)

Essentially, there are three criteria on the time-energy relationship but meanings ofΔ𝑡 are different. About Margolus and Levitin’s paper and Anandan and Aharonov’spaper, this situation is that a certain state moves to the orthogonal state, i.e., we cantake two states with perfect distinguishable. Its minimum time is denoted as Δ𝑡. AboutMandelstam and Tamm’s paper, this situation is that a certain state moves to the statesuch that we can two states with distinguishable for a given observable. Note that thesesituations restrict the Hamiltonian each other.

Bibliography

[1] E. S. Abers and B. W. Lee, Phys. Rep. 9, 1 (1973).

[2] Y. Aharonov, D. Z. Albert, and L. Vaidman, Phys. Rev. Lett. 60, 1351 (1988).

[3] Y. Aharonov, P. G. Bergmann, and J. L. Lebowitz, Phys. Rev. 134, B1410 (1964).

[4] Y. Aharonov and D. Bohm, Phys. Rev. 122, 1649 (1961).

[5] Y. Aharonov and A. Botero, Phys. Rev. A 72, 052111 (2005).

[6] Y. Aharonov, S. Popescu, D. Rohrlich, and L. Vaidman, Phys. Rev. A 48, 4084(1993).

[7] Y. Aharonov, S. Popescu, J. Tollaksen, and L. Vaidman, arXiv:0712.0320.

[8] Y. Aharonov and D. Rohrlich, Quantum Paradoxes (Wiley-VCH, Weibheim, 2005).

[9] Y. Aharonov and J. Tollaksen, to appear in Visions of Discovery: New Light onPhysics Cosmology and Consciousness, edited by R. Y. Chiao, M. L. Cohen, A.J. Leggett, W. D. Phillips, and C. L. Harper, Jr. (Cambridge University Press,Cambridge, in press), arXiv:0706.1232.

[10] Y. Aharonov and L. Vaidman, Phys. Rev. A 41, 11 (1990).

[11] Y. Aharonov and L. Vaidman, J. Phys. A 24, 2315 (1991).

[12] Y. Aharonov and L. Vaidman, in Time in Quantum Mechanics, Vol. 1, edited by J.G. Muga, R. Sala Mayato, and I. L. Egusquiza (Springer, Berlin Heidelberg, 2008)pp. 399-447.

[13] N. I. Akhiezer and I. M. Glazman, Theory of Linear Operators in Hilbert Space(Dover, New York, 1993).

[14] J. Anandan and Y. Aharonov, Phys. Rev. Lett. 65, 1697 (1990).

[15] A. Arai and Y. Matsuzawa, Lett. Math. Phys. 83, 201 (2008).

21

22 BIBLIOGRAPHY

[16] M. Arndt, O. Nairz, J. Vos-Andreae, C. Keller, G. van der Zouw, and A. Zeilinger,Nature 401, 680 (1999).

[17] A. D. Baute, I. L. Egusquiza, J. G. Muga, and R. Sala Mayato, Phys. Rev. A 61,052111 (2000).

[18] C. M. Bender, Rep. Prog. Phys. 70, 947 (2007).

[19] N. Bohr, Nature 121, 580 (1928).

[20] N. Bohr, in Albert Einstein: Philosopher-Scientist, edited by P. A. Schilpp (Cam-bridge Univ. Press, Cambridge, 1949). pp.199-242.

[21] G. Bonneau, J. Faraut and G. Valent, Am. J. Phys. 69, 322 (2001).

[22] V. B. Braginsky and F. Y. Khalili, Quantum Measurement (Cambridge UniversityPress, Cambridge, 1992).

[23] N. Brunner, A. Acın, D. Collins, N. Gisin, and V. Scarani, Phys. Rev. Lett. 91,180402 (2003).

[24] M. Brune, J. Bernu, C. Guerlin, S. Deleglise, C. Sayrin, S. Gleyzes, S. Kuhr, I.Dotsenko, J. M. Raimond, and S. Haroche, arXiv:0809.1511.

[25] P. Busch, P. Mittelstaedt, and P. J. Lahti, Quantum Theory of Measurement(Springer-Verlag, Berlin, 1991).

[26] K. M. Case, Phys. Rev. 80, 797 (1950).

[27] C. Caves, Rev. Mod. Phys. 52, 341 (1980).

[28] C. Caves, Phys. Rev. Lett. 54, 2465 (1985).

[29] L. Childress, M. V. Gurudev Dutt, J. M. Taylor, A. S. Zibrov, F. Jelezko, J.Wrachtrup, P. R. Hemmer, and M. D. Lukin, Science 314, 281 (2006).

[30] T. E. Clark, R. Menioff and D. H. Sharp, Phys. Rev. D 22, 3012 (1980).

[31] G. M. D’Ariano, to appear in Philosophy of Quantum Information and Entangle-ment edited by A. Bokulich and G. Jaeger (Cambridge University Press, Cambridge)arXiv:0807.4383.

[32] B. Davies, Integral Transforms and their Applications (Springer-Verlag, New York,1978).

[33] E. B. Davies and J. T. Lewis, Commun. Math. Phys. 17, 239 (1970).

BIBLIOGRAPHY 23

[34] D. Deutsch, Proc. R. Soc. Lond. A 400, 97 (1985).

[35] D. Deutsch, talk in ”UK-Workshop 2009 on Quantum Information” at British Em-bassy in Japan on 1.22.2009.

[36] P. A. M. Dirac, Proc. Roy. Soc. A 113, 621 (1926).

[37] M. V. Gurudev Dutt, L. Childress, L. Jiang, E. Togan, J. Maze, F. Jelezko, A. S.Zibrov, P. R. Hemmer, and M. D. Lukin, Science 316, 1312 (2007).

[38] A. Einstein, Ann. Phys. (in Berlin) 360, 241 (1918).

[39] A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev. 47, 777 (1935).

[40] E. Farhi and S. Gutmann, Int. J. Mod. Phys. A 5, 3029-3051 (1990).

[41] R. P. Feynman, Int. J. of Theor. Phys. 21, 467 (1982).

[42] T. Fulop, T. Cheon and I. Tsutsui, Phys. Rev. A 66, 052102 (2002).

[43] T. Fulop, Ph.D. thesis, University of Tokyo, 2005.

[44] T. Fulop, arXiv:0708.0866.

[45] A. Furuta, talk in ”Recent Issues and Prospects on Quantum Theory” at KEK, on3.22.2008.

[46] C. F. Gauss, Theoria Combinationis Observationum Erroribus Minimis Obnoxiae(Apud Henricum Dieterich, Gottingae, 1823).

[47] P.-L. Giscard, arXiv:0901.0333.

[48] S. Gleyzes, S. Kuhr, C. Guerlin, J. Bernu, S. Deleglise, U. B. Hoff, M. Brune, J. M.Raimond, and S. Haroche, Nature 446, 297 (2007).

[49] A. N. Gordeyev and S. C. Chhajlany, J. Phys. A 30, 6893 (1997).

[50] M. J. W. Hall, Phys. Rev. A 64, 052103 (2001).

[51] M. J. W. Hall, Phys. Rev. A 69, 052113 (2004).

[52] R. Hanson et al., Science 320, 352 (2008).

[53] S. Haroche and J,-M. Raimond, Exploring the Quantum: Atoms, Cavities, andPhotons (Oxford Univ. Press, Oxford, 2006).

[54] M. Hayashi and F. Sakaguchi, J. Phys. A 33, 7793 (2000).

24 BIBLIOGRAPHY

[55] W. Heisenberg, Z. Phys. 43, 172 (1927).

[56] W. Heisenberg, The Physical Principles of the Quantum Theory (University ofChicago Press, Chicago, 1930; Dover, New York, 1949, 1967).

[57] C. W. Helstrom, Int. J. Theor. Phys. 11, 357 (1974).

[58] P. A. Hiskett, D. Rosenberg, C. G. Peterson, R. J. Hughes, S. Nam, A. E. Lita, A.J. Miller, and J. E. Nordholt, New J. Phys. 8 193 (2006).

[59] F. Hiroshima, S. Kuribayashi, and Y. Matsuzawa, arXiv:0810.2350.

[60] H. F. Hofmann, arXiv:0805.0029.

[61] A. S. Holevo, Rep. Math. Phys. 13, 379 (1978).

[62] A. S. Holevo, Rep. Math. Phys. 16, 385 (1979).

[63] A. S. Holevo, Probabilistic and statistical aspects of quantum theory (North-Holland,Amsterdam, 1982).

[64] A. S. Holevo, Statistical Structure of Quantum Theory (Springer, Berlin, 2001).

[65] O. Hosten and P. Kwiat, Science 319, 787 (2008).

[66] M. Hotta and M. Ozawa, Phys. Rev. A 70, 022327 (2004).

[67] R. A. Hulse and J. H. Taylor, Astrophysical Journal 195, L51 (1975).

[68] C. J. Isham, Lectures on Quantum Theory – Mathematical and Structural Founda-tions – (Imperial College Press, London, 1995).

[69] S. Ishii, Heisenberg’s microscopy (Nikkei Business Publication, Tokyo, 2006) (inJapanese).

[70] S. Ishikawa, Int. J. Theor. Phys. 30, 401 (1991).

[71] S. Ishikawa, Rep. Math. Phys. 29, 257 (1992).

[72] K. Ito, An Introduction to Probability Theory (Cambridge University Press, Cam-bridge, 1978).

[73] F. Jelezko et al., Phys. Rev. Lett. 92, 076401 (2004).

[74] L. M. Johansen, Phys. Lett. A 322, 298 (2004).

[75] P. Jordan, Z. Phys. 37, 383 (1926).

BIBLIOGRAPHY 25

[76] R. Jozsa, Phys. Rev. A 76, 044103 (2007).

[77] S. Kagami, Master Thesis at Tokyo Institute of Technology (2009).

[78] E. H. Kennard, Z. Phys. 44, 326 (1927).

[79] N. Konno, Quantum Inf. Process. 1, 345 (2002).

[80] N. Konno, J. Math. Soc. Jpn. 57, 1179 (2005).

[81] N. Konno, Mathematics on Quantum Walk (Sangyo Tosho, Tokyo, 2008) (inJapanese).

[82] K. Koshino and A. Shimizu, Phys. Rep. 412 191 (2005).

[83] A. M. Krall, J. Diff. Eq. 45, 128 (1982).

[84] K. Kraus, Ann. Phys. 64, 311 (1971).

[85] L. D. Landau and E. M. Lifschitz, Quantum Mechanics Non-Relativistic Theory(Pergamon Press, 1977) Third edition.

[86] U. Larsen, J. Phys. A 23, 1041 (1990).

[87] J. Maddox, Nature 331, 559 (1988).

[88] L. Mandelstam and I. Tamm, J. Phys.(USSR) 9, 249 (1945).

[89] N. Margolus and L. B. Levitin, Physica D 120, 188 (1998).

[90] K. M𝜙lmer, Phys. Lett. A 292, 151 (2001).

[91] G. Mitchison, R. Jozsa, and S. Popescu, Phys. Rev. A 76, 062105 (2007).

[92] J. von Neumann, Math. Ann. 102, 49 (1929).

[93] J. von Neumann, Mathematische Grundlagen der Quantumechanik (Springer,Berlin, 1932), [ Eng. trans. by R. T. Beyer, Mathematical foundations of quantummechanics (Princeton University Press, Princeton, 1955). ]

[94] M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information(Cambridge University Press, Cambridge, 2000).

[95] I. Ojima, in Proceeding of ”Quantum Bio-Informatics – From Quantum Informationto Bio-Information –” at Tokyo University of Science on 3.14-17.2007. pp. 217–228.arXiv:0705.2945.

26 BIBLIOGRAPHY

[96] M. Onoda, S. Murakami, and N. Nagaosa, 93, 083901 (2004).

[97] M. Ozawa, J. Math. Phys. 25, 79-87 (1984).

[98] M. Ozawa, Phys. Rev. Lett. 60, 385 (1988).

[99] M. Ozawa, in Squeezed and Nonclassical Light, edited by P. Tombesi and E. R. Pike,(Plenum, New York, 1989), pp. 263- 286.

[100] M. Ozawa, in Quantum Aspects of Optical Communications (Springer, Heidelberg,1991) pp. 1-17.

[101] M. Ozawa, Phys. Rev. A 67, 042105 (2003).

[102] M. Ozawa, J. Quant. Inf. 1, 569 (2003).

[103] M. Ozawa, Ann. Phys. 311, 350-416 (2004).

[104] M. Ozawa, J. Opt. B Quantum Semiclass. Opt. 7, S672 (2005).

[105] A. Pais, ”Subtle is the Lord... The Science and Life of Albert Einstein” (OxfordUniversity Press, Oxford, 1982).

[106] A. D. Parks, D. W. Cullin, and D. C. Stoudt, Proc. R. Soc. Lond. A 454, 2997(1998).

[107] A. D. Parks, J. Phys. A 41, 335305 (2008).

[108] L. Pauling and E. B. Wilson, Introduction to Quantum Mechanics (Mcgraw-Hill,New York, 1935).

[109] A. Peres and P. Scudo, J. Mod. Opt. 49, 1235 (2002).

[110] G. J. Pryde, J. L. O’Brien, A. G. White, T. C. Ralph, and H. M. Wiseman, Phys.Rev. Lett. 94, 220405 (2005).

[111] M. Redei and S. J. Summers, Stud. Hist. Philos. Sci. B 38, 390 (2007).

[112] M. Reed and B. Simon, Methods of Modern Mathematical Physics II, Fourier Anal-ysis, Self-Adjointness (Academic Press, New York, 1975).

[113] F. Rellich, Math. Ann. 122, 343-368 (1950).

[114] K. J. Resch, J. S. Lundeen, and A. M. Steinberg, Phys. Lett. A 324, 125 (2004).

[115] K. J. Resch, Science 319, 733 (2008).

BIBLIOGRAPHY 27

[116] B. Reznik and Y. Aharonov, Phys. Rev. A 52, 2538 (1995).

[117] N. W. M. Ritchie, J. G. Story, and R. G. Hulet, Phys. Rev. Lett. 66, 1107 (1991).

[118] H. P. Robertson, Phys. Rev. 34, 163-164 (1929).

[119] D. Rohrlich and Y. Aharonov, Phys. Rev. A 66, 042102 (2002).

[120] R. A. Sack, Phil. Mag. 3, 497 (1958).

[121] T. Sagawa, Master Thesis, Tokyo Institute of Technology, 2008.

[122] H. Salecker and E. P. Wiger, Phys. Rev. 109, 571 (1958).

[123] L. I. Schiff, Quantum Mechanics (McGraw-Hill, 1965) Third edition.

[124] E. Schrodinger, Sitz. Ber. Preuss. Akad. Wiss., Phys.-Math. KI. (Berlin), 296 (1930).

[125] A. Shapere and F. Wilczek, Geometric Phases in Physics (World Scientific, Singa-pore, 1988).

[126] Y. Shikano and A. Hosoya, J. Math. Phys. 49, 052104 (2008).

[127] Y. Shikano and A. Hosoya, arXiv:0812.4502.

[128] E. Sjoqvist, Phys. Lett. A 359, 187 (2006).

[129] E. Sjoqvist, Acta Phys. Hung. B 26, 195 (2006).

[130] A. M. Steinberg, Phys. Rev. Lett. 74, 2405 (1995).

[131] A. Tanaka, Phys. Lett. A 297 307 (2002).

[132] A. Tonomura, J. Endo, T. Matsuda, T. Kawasaki, and H. Ezawa, Am. J. Phys. 57,117 (1989).

[133] I. Tsutsui, T. Fulop and T. Cheon, J. Phys. A 36, 275-287 (2003).

[134] J. Twamley and G. J. Milburn, New J. of Phys. 8, 328 (2006).

[135] A. Uhlmann, Rep. Math. Phys. 24, 229 (1986).

[136] A. Uhlmann, Lett. Math. Phys. 21, 229 (1991).

[137] A. Uhlmann, in Proceedings of the First Max Born Symposium, edited by R. Giel-erak, J. Lukierski, and Z. Popowicz (Kluwer Academics Press, Netherlands, 1992)pp. 267-274. mp-arc:92-44.

28 BIBLIOGRAPHY

[138] L. Vaidman, Am. J. Phys. 60, 183 (1992).

[139] L. Vaidman, Found. Phys. 26, 895 (1996).

[140] M. S. Wang, Phys. Rev. Lett. 79, 3319 (1997).

[141] H. Weyl, Math. Ann. 68, 220-269 (1910).

[142] H. Weyl, Gryppentheorie und Quantenmechanik (S. Hirzel, Leipzig, 1928), [ Eng.Trans. H. P. Robertson, The Theory of Groups and Quantum Mechanics (Dover,New York, 1950). ].

[143] N. S. Williams and A. N. Jordan, Phys. Rev. Lett. 100, 026804 (2008).

[144] H. M. Wiseman, Phys. Rev. A 65, 032111 (2002).

[145] H. P. Yuen, Phys. Rev. Lett. 51, 719 (1983).

[146] K. Yokota, T. Yamamoto, M. Koashi, and N. Imoto, arXiv:0811.1625.

[147] ”The Quantum and the Foundations of Physics Conference” at the University ofTexas organized by J. A. Wheeler on 12.17-20.1982.

master thesis 2009 weak values in quantum …shikano/uncertainty_principle.pdfmaster thesis 2009...

Documents